TEACHING PRESCHOOLERS ABOUT CARDINAL NUMBERS

TEACHING PRESCHOOLERS

ABOUT CARDINAL NUMBERS

By Sallee Beneke

Preschoolers¡¯ Cardinal Number Competencies

Before they can become skilled counters, children

master several foundational concepts and skills called

the number core (National Research Council, 2009).

These foundational concepts and skills are cardinality,

the number word list, 1-to-1 correspondences, and

written number symbols.

Numbers are an abstract concept. We use numbers to

describe the quantity in a set (e.g., 3 cookies, 5 chairs, 1

dog). Children come to preschool with some potentially

limited understanding of the relationship between numbers and things that can be done with them (i.e., number

sense). Their competencies grow as teachers and family

members model how to count to determine quantity.

This includes describing the process of counting sets of

objects, providing guidance in using cardinal number to

solve problems and feedback on children¡¯s efforts to use

cardinal numbers. A range of opportunities to develop

cardinal number can be provided through consideration

of the materials available in the environment, adult-child

and child-child interactions, routines, as well as planned

activities. The frequency of opportunities a child has

to develop number competencies is also a factor. When

young children become competent at using cardinal

numbers, they can automatically use the skill in everyday situations. Therefore, it is important for teachers to

provide children with an abundance of opportunities to

practice using cardinal numbers in a variety of situations

and with a range of materials.

Cardinality is the knowledge of how many things are in

a set and the number name for that quantity. Children

who have mastered this competency understand that

the last number counted is the number of objects in the

set. For example, they can accurately count 10 objects,

and when asked how many are in the set, they answer

¡°ten.¡± Preschoolers typically recognize the number of

objects in a set of three or fewer instantly. This ability is

called perceptual subitizing. As they become more familiar

with sets of numbers they begin to see the quantity in

larger sets by quickly recognizing the smaller sets that

make up the quantity (e.g., the set of eight on a die is

made up of two groups of four). This is called conceptual

subitizing (Clements, 1999). For example, children can see

a group of dots on the side of a die and they ¡°just know¡±

how many dots are present without counting them.

Children¡¯s developing competence in understanding

and using cardinality is interrelated with 1-to-1 correspondence and the number word list. For example,

when a child can move her finger down a row of objects

and touch each object in the row just once before moving her finger to the next object, she can then begin to

attach a number name to each object. As children learn

to recite the number word list, she will be better able to

count with accuracy. Development in one core skill or

concept is likely to positively influence development in

another area. Experts in the field of early mathematics

have described developmental pathways or trajectories

for each of the core areas (National Research Council,

2009; Clements & Sarama, 2014). This developmental

view of cardinality is summarized in Table 1.

Table 1. Steps/Ages in Learning to Think About Cardinality*

Steps/Ages

Skill

Related Competencies

Step 1:

Beginning Two & Three Year Olds

1.1 Subitizing

Uses perceptual subitizing to give the number for 1, 2, or 3 things.

Step 2:

Later Two & Three Year Olds

2.1 Subitizing

Continues to generalize perceptual subitizing to new configurations

and extends to some instances of conceptual subitizing for 4 and 5.

Can give number for 1 to 5 things

STEP 3:

Four Year Olds

3.1 Using 5-groups

Extends conceptual subitizing to groups of 5 plus 1, 2, 3, 4, 5 to see 6 though 10

(e.g., 6 is 5 + 1, 7 is 5 + 2).

3.2 Using fingers

Uses 5 fingers on one hand plus additional fingers on the other hand as a kinesthetic

and visual aid for conceptual subitizing.

*Adapted from National Research Council (2009)

Strategies for Helping

Preschoolers Learn About Cardinality

Engaging young children in the following five mathematical processes helps them develop and communicate

their thinking about all areas of mathematics, including

cardinality (National Council of Teachers of Mathematics,

2000). These mathematical processes are: (a) representing, (b) problem solving, (c) reasoning, (d) connecting,

and (e) communicating. Educators can teach children to

use these five processes to mathematize or relate shape

concepts to their everyday world. Tables 2 and 3 provide

examples of language and materials that teachers can

employ to help children use these processes.

Representing. Children may represent their understanding of number in a variety of ways. For example,

children might count out five crackers for each of their

friends or count the characters as they arrive in the

book, The Doorbell Rang. Teachers can encourage children

to represent their understanding of the quantity in a set

by drawing it, or by representing it with other objects.

For example, if a child says that there are five people

in his family, the teacher can ask the child to represent

them with counters (e.g., ¡°Can you show me with buttons?¡±), or children can draw each member of a set. For

example, if a child says that he played with three friends

on the playground, the teacher can ask him to draw the

group of friends. Preschoolers also love to use fingers

to represent sets of people or objects. When asked how

many people are in her family, a child may hold up four

fingers, wiggling each finger as she describes the family

member (e.g., ¡°mommy, daddy, my brother, and me¡±).

Problem solving. ¡°Problem solving and reasoning are the

heart of mathematics¡± (NAEYC, 2010). Young children

learn by engaging with and solving meaningful problems in their everyday environments. Young children

love conducting surveys to find answers to questions

that they pose. For example, one four-year-old labeled

one column of a T-chart with a drawing of a sun and the

other column with a picture of the sun with an X drawn

over it. He went from child to child asking, ¡°Do you like

the sun?¡± He made a tally mark in the column that

matched each child¡¯s preferences. After he had gathered

Engaging young children

in five important

mathematical processes

helps them develop

and communicate

their thinking about all

areas of mathematics,

including geometry

data from several children the boy counted the tally

marks in each of the two columns and wrote the number that represented the quantity at the bottom of each

column. Group games that involve counting challenge

children to apply their counting skills. For example, a

bowling game with plastic pins challenges children to

count the number of pins that are knocked down and

the number that remain standing (Charlesworth, 2012).

Board games that make use of dice and game pieces

that move along a path are enjoyable to young children

and challenge them to figure out how many spaces they

can move their game pieces along the path.

Reasoning and proof. Teachers can challenge a preschooler¡¯s reasoning by conversing with him about his

work with quantities and asking him to explain how

he came to a certain conclusion about the quantity

represented by a set (e.g., ¡°How do you know there are

six children at the round table?¡±). The child will typically

recount, demonstrating how he reached his conclusion using rational counting skills. Children also can

learn to demonstrate the accuracy of their conclusions

by representing the objects in the set with fingers or

counters. For example, if a child says he has four pieces

of candy (3 candy canes and 1 Tootsie Roll), the child can

count out three red counters for the candy canes and

one black counter for the Tootsie Roll. The child can then

count the total number of counters (both red and black).

Connecting. Teachers can help preschoolers see the

connection between rational counting and their everyday world as they naturally occur (e.g., ¡°Look, there

are horses by the fence! How many do you think are

there?¡±). Routine activities, such as snack time provide

many opportunities for children to see the value in

figuring out the answer to the question, ¡°how many?¡±

For example, children can count the number of children

in attendance and then use that quantity to figure out

how many snacks, cups, napkins, etc. to set out.

Communicating. Encouraging children to communicate their thinking by verbalizing, drawing, writing,

gesturing, and using concrete objects or symbols can

help them share their ideas about quantity with other

children and adults. As children learn to count larger

sets, adults can challenge them to apply this ability in

everyday contexts and to explain how they determined

the quantity. Adults can help children learn mathematical terms related to cardinal number, such as ¡°quantity,¡±

and ¡°set¡± by modeling them in conversations (e.g., ¡°Can

you please put six muffins on the plate¡±).

Strategies for Supporting Dual Language Learners

Several strategies can be used to help Dual Language

Learners (DLLs) learn about cardinal numbers. The

teacher can refer to quantities of objects in the young

DLL¡¯s home environment as she engages him/her in

informal conversations (e.g., ¡°You have three perros/

dogs at your house. Let¡¯s count how many stuffed perros/dogs we have in our cozy corner.¡±). The teacher can

gather the background information needed for this type

of conversation by establishing a friendly, collaborative

relationship with the young DLL¡¯s family, conducting

informal interviews with them about the child¡¯s home

life, and/or making one or more visits to the young DLL¡¯s

home. It is most effective for the young DLL to learn the

number word list (i.e., one, two, three, four, five, etc.)

first in his/her home language, and then the teacher

can help the child learn to count in English. Since the

English number names (i.e., one, two, three, four, five,

etc.) typically do not share cognates or linguistic roots

with other languages, it will likely take a great deal

of practice for the young DLL to associate the English

number list with that of his/her home language. The

teacher can help the child begin to associate the number list with quantities by using props and gestures.

Fingers are especially helpful when teaching cardinal

numbers. For example, the teacher can count the child¡¯s

fingers first in the DLL¡¯s home language and then in

English (e.g., ¡°How many fingers am I holding up? Let¡¯s

count them in espa?ol! Uno, dos, tres, cuatro, cinco, seis.

Now let¡¯s count them in English. One, two, three, four,

five!¡±). Visual cues in the environment also can support

the young DLL¡¯s understanding of cardinal numbers. For

example, a colorful, attractive number list with pictorial

representations and the number names in English and

Spanish can be referenced by the young DLL, peers, and

teachers, and family members. The teacher also can

support a young DLL¡¯s mastery of cardinal numbers by

selecting one or more high quality children¡¯s books that

are written in English and the child¡¯s home language

(e.g., Mouse Count/Cuenta de rat¨®n by Ellen Stoll Walsh).

Repeated dialogic readings in small and large group

can allow the teacher, young DLL, and peers to discuss

cardinal numbers, while referring to examples in the

book. For further information, see the microteach guide,

Supporting Mathematical Learning of Young Dual Language

Learners (Beneke, 2016).

Table 2. Examples of teacher language that supports

children¡¯s mathematical processes* with cardinality

Table 3. Examples of useful materials for teaching

and learning about cardinality in preschool

Representing

Blocks

How many are there?

Unit blocks

Can you show me how many there are?

Table-top blocks

Let¡¯s draw a picture that shows how many are in the set.

Can you show me how many with your fingers?

Legos?

Table Toys

How can we use these counters to show how many are in the set?

Spinners

Dice

Problem-Solving

Path games

Can you use the counters to keep track of how many we have?

Counters

Can you count them to see how many we will need?

Linking chains

I wonder how we can figure out how many we will need?

Pop beads

How many are there? How do you know?

Unifix cubes

How many do we need?

Ten-frames

Reasoning & Proof

Boards

How do you know we need that many?

What makes you think there are

Flannel-board sets

of them?

Can you make a mark on your paper for each one? Then we¡¯ll count them.

Path games that require subitizing and moving a game piece

along a path

What if the bowl is empty? How many will we have, then?

Connect Four and other games that encourage counting sets of items

How did you know there were three dots on the side of the die?

You didn¡¯t count them!

Books

There are so many¡ªhow will we figure out how many there are?

12 Ways to Get to 11 by Eve Maerriam

Connecting

Anno¡¯s Counting Book by Mitsumasa Anno

Fish Eyes by Lois Ehlert

How far can you count? Do you think there are that many in our pile?

How can we find out?

Which pile has more?

How can we give everyone the same amount?

Why are you touching them when you count?

Communicating

Miss Julie says there are seven babies in the housekeeping area.

Is she right?

There are lots of oranges in this bowl¡ªhow can we count them?

Should we take them out?

How can we arrange these stickers so we can count them better? Why?

We had five, and I just found another one. Now how many do we have?

There are fifteen children here today. How many napkins will

we need for snack?

*Mathematical processes described by the National Research Council (2009).

Mouse Count by Ellen Stoll Walsh

Ten Black Dots by Donald Crews

The Button Box by Margarette Reid

Press Here by Harve Tulletth

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